Information Technology Reference
In-Depth Information
So how do we model this effective synaptic strength? As Feng [
29
] emphasizes, phys-
iologic data clearly show that nearby neurons usually fire in a correlated way [
72
]
and anatomical data reveal that neurons with similar functions group together and fire
together [
56
]. The SHM proposed by West and Grigolini [
69
] represents the output from
the dynamics of a cluster of neurons, that is, from a tightly coupled network of parallel
channels. The synaptic strength has contributions from each of the channels in a given
time interval, with the time constants contributing to
changing randomly from
channel to channel. If such a cluster were modeled by a collection of two-state elements
that strongly interact with one another, the dynamics of individual elements synchro-
nize [
64
,
68
] and the global behavior would be characterized by an extended sojourn in
one of the two states, with abrupt random jumps from one state to the other, as shown
earlier. The distribution density of sojourn times, or equivalently the distribution den-
sity of switching between states,
w
eff
(
t
)
ψ(
t
)
, is a consequence of the network dynamics. In
the present case of multiple channels
ψ(
t
)
is determined by the distribution of time
constants.
The formal analysis determining the response of a physical web S to a weak pertur-
bation was addressed by a number of investigators [
45
,
59
]. These investigators studied
the response of a complex web characterized by the production of non-ergodic renewal
events to a simple harmonic perturbation of weak intensity and established that the web
response to the harmonic perturbation is transient and asymptotically fades to zero, as
we discussed previously. To make use of this result in the context of habituation we
cannot rely on the traditional LRT, based as it is on the assumption that the web S is
stationary. The property of stationarity is often violated by complex webs such as the
neuronal networks considered here. To properly address the issue of information trans-
mission within a complex web we adopt the generalized LRT [
4
,
7
]. We assume that the
dynamics of interest are described by the event-dominated fluctuations
ξ
s
(
)
generated
in the complex web and that the time dependence of the perturbation is described by the
function
t
. We assume that the latter is independent of the former, whereas when we
switch on the interaction between the web and the environment the fluctuations
ξ
p
(
t
)
ξ
s
(
t
)
can
be perturbed by
. In the absence of perturbation the average over the ensemble of
realizations of the web of interest vanishes, in which case the most general form of LRT
is given by
ξ
p
(
t
)
t
0
χ(
t
)ξ
p
(
t
)
dt
,
ξ
s
(
)
≡
t
t
,
(7.172)
t
)
where
ξ
p
(
t
)
is the stimulus. The quantity
χ(
t
,
is the linear response function of the
web S and can be written
t
)
dt
,
d
(
t
,
t
)
=−
χ(
t
,
(7.173)
t
)
where
(
t
,
is the aged survival probability defined by
t
)
=
ξ
s
(
t
)
.
(
t
,
t
)ξ
s
(
(7.174)
